Page 400 - Cam Design Handbook

THB12 9/19/03 7:34 PM Page 388

388 CAM DESIGN HANDBOOK

1 ds
tana = .
R q () d q
Thus the nominal follower velocity is
˙ s = w R( tan)a (12.21)

where w is the constant input angular velocity of the shaft. Integrating Eq. (12.21) gives
the nominal displacement at the roller
t
s = w R() q tan a dt. (12.22)
Ú 0
The equivalent stiffness k eq of the system is derived in Koster (1975).
1 1 1
2
= + tan a (12.23)
k k k
eq c t
where k eq, the equivalent vertical spring stiffness, is
kk
k = x
c
+
kk
x
and k c, the equivalent tangential spring stiffness, is
kk
k = y q .
c
k + k
y q
k eq thus obtained is time-dependent.
The original model with four degrees of freedom may be approximated by a model
with a single degree of freedom with variable stiffness, using the general application for
the rule of transformation. The equation of motion of the system
˙˙ + (
˙ -
-
mx c s x ˙)+ k ( s x) = 0
eq
where the dots represent differentiation with respect to time.
Using Eq. (12.23) gives
k
-
˙ -
mx c s x ˙)+ c ( sx) = 0 . (12.24)
˙˙ + (
k Ê 1 ˆ 2
1 + c s ˙
k t Ë w R ¯
The effect of the flexibility of the shaft is greatest for the position of the cam correspon-
ding to a = a max. For many cam profiles this occurs at the midpoint of the cam. As a first
approximation, R(q) can be replaced by the mean pitch radius of the cam
R q () = R . (12.25)
m
Eq. (12.24) can now be rewritten as
k
mx c s x ˙)+ c 2 ( sx) = 0 . (12.26)
-
˙ -
˙˙ + (
k Ê 1 ˆ
1 + c Á s ˙˜
k Ë w R m ¯
1
Next, the following nondimensional parameters are introduced: